# Questions tagged [monomorphisms]

For questions related to monomorphisms, which are categorical generalizations of injective functions.

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• 1,649
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### About the universal morphism from the pushout of monomorphisms.

Let $A,B,C$ be objects in a category $\mathrm{C}$ and we have the pushout diagram of monomorphisms$\require{AMScd}$ \begin{CD} C @>>> A\\ @VVV @VVV\\ B @>>> A\bigsqcup\limits_C B\end{...
• 21
1 vote
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### Show that if $G$ is the right adjoint of restriction-of-scalars along any ring homomorphisms $A\to B$, then $G$ is always a monomorphism

Let $A$ and $B$ be rings. Show that the right adjoint of restriction-of-scalars functor along any ring homomorphism $f: A \to B$ preserves injectives and show that the unit of this adjunction is ...
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• 23
1 vote
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### If $f:Y \to X$ is a mono in $\textsf{Set}^{\mathbb{C}^{op}}$, then $f$ factors uniquely through $Y \xrightarrow{g} A \stackrel{i}{\hookrightarrow} X$.

I have taken an introductory course in category theory and would like to learn more about presheaves. Currently I am working through "Generic figures and their glueings" by Marie La Palme ...
• 371
1 vote
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### Does subclass of morphisms closed under wide pullbacks necessarily consist of only monomorphisms?

For category $\mathcal{C}$, does subclass $M \subseteq \operatorname{Mor}(\mathcal{C})$ closed under even largely wide pullbacks necessarily consist of only monomorphisms? The assumption being more ...
• 75
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### If for monos $u\leq v$ and $v\leq u$ then their domains are isomorphic

I'm unable to prove that if a mono $u:B\to A$ is less then mono $v:C\to A$ by $f:B\to C$ with $v\circ f=u$ and also $v\leq u$ by $u\circ g=v$ then $B\cong C$ by using some morphisms as above, but I ...
• 3,978
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### Category where not every mono is regular

I'm looking for a category with the titular property. Since $f : A \to B$ is supposed to be mono, the desired factorisation is automatically unique (correct?). Thus, I would need to find a category in ...
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• 12.1k
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### In general, how does one show that a mapping between two groups is a monomorphism? Example included.

The definition of a group monomorphism is as follows: Let $G$ and $H$ be groups and suppose $\phi:G\to H$ is a group homomorphism. Then $\phi$ is a group monomorphism if and only if $\phi$ is an ...
• 649
1 vote
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### Doubt proving that $(\forall f,f' \quad g\circ f = g\circ f' \implies f=f' ) \implies g \text{ is injective}$?

I am trying to understand the proof of: (ii) The function $g : b \to c$ is mono iff the following condition holds: for any two functions $f,f' : a \to b$, $g \circ f = g \circ f'$ implies $f=f'$ ...
• 24.2k
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### Categories with colimit-stable monomorphisms

In Definition 2.12 of https://arxiv.org/pdf/1409.3805.pdf, Adamek defines the notion of a cocomplete category having stable monomorphisms, meaning that small coproducts of monomorphisms are ...
• 1,621
1 vote
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### $E(M)$ is indecomposible if and only if $M$ is uniform.

I tried to prove the above mentioned statement written in the book "Serial Rings" by Gennadi Puninski. For the first part I assumed that the module $E(M)$ is indecomposable. I was going by ...
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### Do open continuous maps/local homeomorphisms between locales possess adjoints?

Recently I started learning "theory of locales" (point-free topology) by my-self. While being a very beautiful, natural subject and parallel to point-set topology, some of its notions are ...
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### Confusing Exercise in Hilton/Stammbach's *A Course in Homological Algebra*

An exercise in Hilton/Stammbach's book begins: "Show that $j : X_0 \to X$ in $\mathfrak{T}$ [the category of topological spaces and continuous maps] is a homeomorphism of $X_0$ into $X$ if and ...
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